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SPEED REGULATION AND PRESSURE VARIATIONS

IN
HYDRO-ELECTRIC PLANTS
WITH

SOLUTIONS OF COMMERCIAL PROBLEMS.
THESIS

, ’ iis:
a, ©
ek
PREF ACE

In the use of formulae for the solution of practical
problems it is generally sufficient for the practising
engineer to check their derivation and correctness but
once, and then apply them wherever an intelligent study
of the subject warrants it. In arranging this article
the above has been kept in mind and the contents have
been separated in such a manner that the part to which
reference would generally be made, after a first careful
study of the whole, -is as concise as possible. That
part is naturally the one giving solutions of practical
problems, and this has been separated from the description
of the problems and the derivation of the formulae. The

derivation of the principal ones of the latter are under-

taken in separate appendixes.

103945
CHAPTER I
CHAPTER II

CHAPTER III

CHAPTER IV

CHAPTER V

CHAPTER VI

CHAPTER VII

APPENDIX A
APPENDIX B
APPENDIX ©

CONP?EW#S

On Speed Regulation in General........ Page

Regulation with Respect to
Fly-Wheel Effect Only ........ccceeeee Page

Regulation with Respect to
Pressure VariationsS -...ccssccccccvces Page

Pressure Variations and Speed
Regulation as Effected by the
Pressure Regulator -..cseccccccccssece Page

Pressure Variations and Speed
Regulation as Effected by the
Equalizing Reservoir or Stand-Pipe.... Page

Pressure Variations and Speed

Regulation as Effected by a

Combination of Stand-Pipe and

Pressure Regulator ....c.scecccccscece Page
The Effect of the Synchronous

By-Pass and the Deflecting Nozsle

upon Speed Regulation ........ccccccce Page
Derivation of Fly-Wheel Formla...... Page
Fly-Wheel Effect eoeeeoeeeeeoeeeeeneeveenee Page

Derivation of Pressure Variation Formila "

il

16

235

38

41
45
53
56
I-l.

CHAPTER I.

ON SPEED REGULATION IN GENERAL.

fhe success of the comparatively recent application
of hydraulic power to the operation of alternators in
parallel, and to the generation of current for electric
lighting, street railway and synchronous motor loads, has
been largely dependent upon the possibility of obtaining
close speed regulation of the generating units accompanied
with good water economy and without undue shock upon
machinery and penstocks while working under extremely
variable loads.

In the present state of governor development, there
is no reason why commercial regulation should not be
obtained in every hydro-electric plant, and should, if not
always equal to, at least be very nearly as satisfactory
as that obtained in modern steam plants. This, however,
can not always be accomplished by the installation of a
well-designed governor alone, as the governor can only
perform certain functions of many which it is necessary to
execute, no matter how perfect the governor. In other
words, it is necessary to put the whole hydro-electric
plant in such a relation of its several parts that the
governor, if it is properly designed, can perform the
function necessary to obtain good commercial regulation.

The ideal governor would effect a change in output
by varying only Q., the quantity of water necessary for a

1
oN
I - 2.

given load, thus obtaining perfect water economy by con-
serving umneeded water for future use. This is not
possible in commercial practice, as head, water, and
therefore efficiency, are usually wasted when operating a
turbine under other than its normal load, and during the
change of load.

The regulation of hydro-electric units as now accon-
plished is one of degree only, since a departure from
normal speed is necessary before the governor can act.
Since the immediate éffect of the gate motion is opposite
to that intended, the speed will depart still further from
the normal. This tends to cause the governor to move the
gate too far, with the result that the speed will not only
return to normal as soon as the inertia of the water and
the rotating parts is overcome, but may rush far beyond
normal in the opposite direction. The obvious tendency
is thus to cause the speed to oscillate above and below
normal (hunting or racing) to the almost complete destruo-
tion of speed regulation.

A successful governor must therefore "anticipate" the
effect of any gate movement. It must move the gate to,
or only slightly beyond, the position which will give
normal speed when readjustment to uniform flow in the
flume or penstock has taken place. A governor with this
property or quality is commonly said to be "dead-beat".
The expedient used for this purpose on hydraulio governors
is called a compensating device, which may be either simple
or compound.

For the regulation of turbines in open flumes, where
the velocity of approach does not exceed 2 to 3 ft. per sec.,
the effect of a gate movement only slightly increases or
decreases the head, so that the immediate effect of the
gate motion may be considered to do what it is wished to
do, namely, increase the flow of water for gate opening
and decrease it for gate closing, so that the inertia of
the water may be disregarded, consideration being necessary
for the inertia of the rotating masses only. Thus in
plants where turbines in open flumes are used, the danger
of hunting or racing is practically eliminated, and almost
any desired speed regulation can be obtained, by choosing
the proper value for the rotating masses. However, this
is true only where the length of the draft tube is short
compared with the head. The water in the draft tube must
be accelerated and retarded at each change of gate opening
and its kinetic energy changed at the expense of the power
output in exactly the same manner as that in the penstock
or flume. The effect is a tendency towards hunting and
racing, and it may be found that regulation with a
governor is absolutely impossible at part load. . Im any
case where a draft tube is used it must be included in all
calculations. If the draft tube is long and the velocity
in it comparatively high, a quick closure of the turbine

gates may cause the water in the draft tube to run away

3
I- 4.

from the wheel, actually creating a vacuum, and then return
again causing a destructive blow against the turbine. In
such a case regulation of any kind is entirely out of the
question.

From what has just been said it is evident that with
turbines in open flume, if the velocity of approach in the
flume and the draft tube are of proper dimensions, the
regulation of the speed is entirely a question of the
inertia of the rotating masses or fly-wheel effect. (See
Appendix B.)

This is the simplest problem in speed regulation and
its solution will be taken up in Chapter II.

If the water is conducted to the turbine in a long
penstock (encased turbine), a large amount of energy is
stored in the moving column of water and a change in its
velocity involves a change in its kinetic energy, which
may, if an attempt is made at too rapid regulation, leave
the turbine deficient in energy when increased power is
desired, or, when the power is decreased, may produce such
shocks as will seriously affect regulation, or perhaps
result in serious injury to the penstock or turbine.

As already stated, the immediate effect of a gate
movement is opposite to that intended, due to this

pressure increase or decrease. With open flume turbines
these pressure variations may be disregarded, but with

encased turbines they must always be carefully considered.

4
 

 
These variations can never be done away with entirely in
practical cases, but they can be minimized to almost any
extent by properly proportioning the various parts of a
plant. The solution of such a problem is generally
determined by commercial considerations, and 2a balance must
be struck between cost of penstocks and cost of fly-wheels,
if it is impossible or undesirable to modify conditions with
other appurtenances.

The solution of this problem will be taken up in
Chapter III.

If in the foregoing a reasonable solution, such as
will satisfy both engineering and commercial conditions,
or either, cannot be found, it will become necessary to
consider one or more practical appurtenences. Eoonornically
it will often prove desireable to consider these, even if
not necessary to a solution of the problem from an engineer-
ing point of view.

A pressure regulator is a device attached to the
turbine casing or penstock near the turbine, and opereted
from the governor in such a manner that when the turbine
gates are closed suddenly and a sufficient amount to
disturb the regulation on account of pressure rise, it will
be positively onened from the governor, sufficiently to
keep the pressure rise within reasonable linits. such
& pressure regulator should be adjustable so that the |

amount of pressure rise for any given load change can be
 
 

 

predetermined for any set of conditions. It should close
automatically and slowly under usual conditions, so that
its sudden closure will not produce @ pressure rise. It
should, however, be so arranged that if, after a sudden
decrease in load, during which the regulator was opened,
a sudden increase should occur, the governor will positively
close it quickly, as the water which would otherwise pass
through the regulator, while slowly closing, is then needed
to prevent an excessive pressure drop. Such pressure
regulators can be designed practically, with all the
functions described, and, if of sufficient size, the
maximum pressure rise will not exceed 10% above normal
with a load change of 100%. It will be seen from our
subsequent discussion that for a fixed set of conditions
the pressure drop is always less than the pressure rise,
which at once shows us the value of the pressure regulator.

The determination of the effect of the pressure
regulator on speed regulation will be considered in
Chapter IV.

As has just been stated above, the pressure drop is
always less than the pressure rise, and, as long as the
pressure drop is within such limits as will permit satis-
factory regulation, the pressure rise can be kept within
safe limits by means of the pressure regulator. When,
however, the point is reached where the pressure drop is

excessive, the problem is again altered. The practical
SS
I- 7.

solution of such a problem is by means of an “equalizing
reservoir" or a “stand-pipe”. The effect of either is

the same, namely, to reduce the effective length of the
penstock, and to supply or take up water during a change

of load, while the flow of the same amount of water would
be accelerated or retarded in that portion of the penstock
beyond the reservoir or stand-pipe. In small plants the
equalizing reservoir is seldom used, as the same effect can
be had cheaper with a stand-pipe. However, in large
plants under medium heads, where the quantity of water used
is considerable, the reservoir will usually prove more
economical in first cost and of much greater benefit to
pressure variations and hence to speed regulation. The
larger the surface area of a reservoir or stand-pipe the
smaller will be the pressure variation, which shows the
advantaze of the reservoir at once. The topography of the
country surrounding a power site usually decides whether a
reservoir or stand-pipe should be used, as entirely
artificial reservoirs may prove very expensive, and in such
cases the stand-pipe is resorted to. It will often be
found advantageous to change the shortest or most economical
route of the pipe line to one more circuitous, in order to
use a natural reservoir site, or to bring either the
reservoir or the stand-pipe closer to the power-house.

If the stand-pipe is of suitable diameter and close to the
turbine, the speed regulation will approach that obtainable

in open flume. Otherwise the problem becomes that of a

7
 
I —- 8.

plant with a closed penstock, of a length equal to that of
the draft-tube, plus the penstock from the turbine to the
stand-pipe, plus the height of the stand-pipe itself.
Stand-pipes to more nearly approach the effect of regulating
reservoirs should have their upper part enlarged, in the
shape of a tank. This tank may be supported on structural
steel columns. The pipe leading to this tank should be of
a diameter not less than that of the penstock. Where a
power house is located near a gently sloping hillside, the
stand-pipe may be laid up this hillside, and supported by
it, instead of being supported by colums. Stand-pipes
for heads of a thousand feet have thus been built.

This problem in treated in Chapter V.

From the foregoing it will be seen that the location
of the stand-pipe or regulating reservoir cannot always be
as desired, and the solution of the problem of pressure
variation is in many cases only partially solved by then.
It is true that an entirely artificial reservoir or a
stand-pipe could be located almost at will, but the cost
would usuelly prove excessive, and, unless a natural site
for either is available, their distance from the power
house will reduce itself largely to a question of cost, and
the cost of increased fly-wheel effect must be balanced
against the cost of closer proximity of stand-pipe to
turbine, to produce the desired regulation.

As already stated in our discussion of the pressure
I= 9.

regulator, the pressure drop is always less than the pressure
rise, so that it is only necessary to bring the stand-pipe
near enough to the turbine so that the pressure drop will

not be excessive, disregarding the pressure rise and pro-
viding for it by means of a pressure regulator.

The problems arising from this combination of pressure
regulator and stand-pipe will be solved under Chapter VI.

If a case should arise which cannot be solved by one
or more of the methods discussed, either on account of
engineering difficulties or due to commercial considerations,
@ synchronous by-pass in the case of the reaction turbine,
or a deflecting nozzle or hood in the case of an impulse
turbine must be resorted to. It may be well to state here
that all of our attempts at close speed regulation are made
with the idea that economy of water is of practically equal
importance to close regulation and that we would regulate
the speed with a governor approaching the ideal, which we
have defined in the first part of this chapter.

Both of the devices mentioned above, and which are too
well known to require description, are extremely wasteful,
and although numerous modifications of these devices have
been made which will reduce their wastefulness, such devices
usually interfere with automatic regulation, and the lessen-
ing of their wastefulness is one of degree only. In places
where fixed quantities of water must be passed, regardless

of the load on the plant, such as in plants where no
I - 10.

storage whatever is available, or where irrigation or other
laws require it, these two methods of regulation may prove
the proper solution.

Their application and effect will be discussed in
Chapter VII.

No solution of a problem in speed regulation is
commercially correct, unless accompanied with an analysis
of the probable load curve of the power plant to be regulated.
It is evidently useless to provide against something which
will never occur. Thus, if a plant consists of a number
of large units, the greatest load change to be considered
should be carefully gone into and, this decided, it should
be known over how many units this must be distributed; in
other words, what will be the least number of units running
in parallel on one circuit. In a large plant it is very
seldom necessary to consider a total load change, or 100%
change of load, whereas in small plants this may frequently

occur, especially when only one unit is running.

LO
II -l.

CHAPTER II.
REGULATION WITH RESPECT TO FLY-WHERL EFFECT ONLY.

In general it may be assumed that where pressure
varistions do not exceed 25% the regulation will be prac-
tically the same as for a turbine in open flume. This
includes turbines with short penstocks and low velocity,
and those cases where the stand-pipe is near enough to the
turbine so that the pressure variations will not exceed the
above.

In practice we may meet the solution of this problem
in various forms, as follows:

Case I. The fly-wneel effect being fixed, to find the
regulation possible and the regulating time.

Case II. The desired regulation being given, to find
the fly-wheel effect and the regulating time.

Case III. The regulation and fly-wheel effect being
given, to find the regulating time necessary.

These three cases can be solved by the single formula

dad « 800,000 x U xt » the derivation and various forms
n& x Wre

of which are given in Appendix A.

For illustration we will solve one or more problems in
each case.

Case I: 1,000 H.P. - 500 K.W. unit under 25 ft. head.

From table of standard runners (See Table X), this
will be a 40" Twin Type F runner turbine, with a 60 cycle
synchronous speed of 180 R.P.M.

11
' JI - 2.

Moment of Inertia of rotating masses (Wr?) « 400,000
ft.2 lbs.
Regulating time (T): If a hydraulic governor is to be
used the regulating time can be chosen at will, but should
not be much less than 2 sec. in any case. In case a
mechanical governor is decided upon, the regulating time
must be obtained from the maker or his catalogue, and the
work required of the governor must be known. (See p. K, App.A).
If we assume a regulating time T «x 2 seo, and substi-

tute in our formla da «2 800,000 x W x f we have

53
a _ 800,000 x 1,000 x 2 0 35". 12.36
180 x 180 x 400,000 '

deff. « 0.8 xd (Type F runner) « 0.6064 w= 0.0745
= 7.45% (See Table II, p. f, App, A.)

 

To find the variation of speed for part load: (See
p. i, App. A); See Figure 1, for diagram. |

For sudden changes of load amounting to 250 H.P. (254),
500 H.P. (50%), 750 H.P. (75%) and 1,000 H.P. (100%), the
speed variation will not exceed 2.75%, sf, 6.75% and 7.45%

respectively.

Case II: 2,500 H.P. unit operating under 20 ft. head.
This would be a quadruple turbine with 55” Type F runners,
with a 25 cycle speed of 125 R.P.WM. Assume the greatest
load change ever expected does not exceed 1,500 H.P., and
this may happen when only one unit is running. The desired
speed regulation for a sudden change of load amounting to
1500 H.P. is 5%.

1l2
II ~ 3.

Regulating time (T) ws» 3 sec.
Fly wheel effect (Wr*) - ?

° d eff.
d eff. - O.8 x a e e qd ¢« 0.8 xX
For a Type F runner 0.8 x = 0.606 .. 4 «& oe
0.05 of
wr - 800,000 BW x Tf 800,000 x 1,500 x 3

 

 

n° a "785 x 125 x 0.0825
2,790,000 f£t.* lbs.

This necessary fly-wheel effect will be composed of:
(a) Generator Rotor:
(bd) Four (4) 55" Type F runners;
(o0) Water in four (4) 55" runners.
(a) may be varied, (b) and (c) are fixed. We will
therefore find (b) and (c) to find the necessary value of (a).
Fach 55" Runner (Type F) weighs 5,850 lbs.

The radius of gyration of Type F runners is approximately ©

O.75 Dia. 0.75 x 55
5 Sx is * 1.72 ft.

We of four runners e« 5,850x4x1.72x1.72 =
69,265 ft.* lbs.

 

The quantity of water in the runners per second may be
found from the formula Q =< HP. x ii » assuming an

H
efficiency of 80%.

o.- Qe 1500 211 se 825 ou. ft. per sec.

Weight = Qy where y = 62.5 w» the weight of one

cubic foot of water.
. . Weight = 825 x 62.5 e 51,560 lbs.

The radius of gyration of the water is practically
13
iI

ij

,
4
II —- 4.

equal to that of the runner = 1.72 ft.

.. Wr? of water = 51,560x1.72x%1.72 =
152,615 ft® lbs.

(>) + (c) = 221,880 ft lbs.

(a) w= 2,790,000 ft* 1bs. — 221,880 ft* lbs. «
2,568,120 £t* 1bs., which is the Wr® that must be obtained
from the generator rotor to obtain no greater variation of
speed than 5% for a sudden load change of 1,500 H.P. on one
unit.

If the load should be such that there would never be
less than two units operating in parallel on the same
circuit, with the greatest load change as before, then the

fly-wheel effect of each generator rotor would only have to

be £2790,000 — (2 x 221,880)

5 = 1,173,120 ft* lbs.

CaseIII: 1,000 H.P. Turbine unit with short penstock,
operating under 50 ft. head.

Ply-wheel effect of generator rotor 600,000 ft® lbs.

Desired regulation: 8% variation for 100% load.

fo find: regulating time, T.

From table of standard runners, this would be a single
turbine, Type E — 424" Rumner at 225 R.P.M. for 60 cycle

synchronous speed.
7" 2 Wr? n* d

(See p.c, App. A).
W x 800,000 Pe Os SPP

Wr? «= 600,000 ft? lbs. W2 w 225 x 225. NW 1,000 H.P

 

a ow a and 0.8 X = 0.645 for Type E runner

ad eff. = -08
14
It — 5.

0.08
eo @ d = 0.645 = 0.124

Substituting, we have # 600,000 x 225 x 225 x 0.124

1,000 x 800,000
= 1.57 seconds.

This closing time is not too quick for a turbine of

the type and size under consideration.

If the regulation is now desired for part loads, we

proceed as under Case I, or as explained in detail in

App. A, p. i.

15
Ill -~ 1.

CHAPTER III.
REGULATION WITH RESPECT TO PRESSURE VARIATIONS.

Pressure variations to a greater or less extent occur
in every practical turbine installation whenever the amount
of water which flows to the turbine is varied, as in the
case of a change of load. However, for practical con-
siderations, as far as regulation of speed with governors
is concerned, these variations can be considerable before
they will have any appreciable effect on the speed regula-
tion. In modern plants, where the tendency usually is to
develop as much power and head as possible in a single plant,
it frequently becomes necessary to carry the water for some
distance in flumes, pipe lines or tunnels, in order to
concentrate the head on the plant. The length of the
conduits often becomes many times greater than the head and,
as such long conduits become a large factor in the cost of
the plant, the natural tendency is to reduce the size of
conduit and increase the velocity of the water flowing in
or through it. However, head, length of conduit, and
velocity of water in the conduit, are the three principal
factors that produce pressure variations, and their relation
to each other, as well as their effect on regulation of
speed, becomes a very important problem in such plants.

In such a plant the effect of the fly-wheel masses is
the same as before, and the same theory and formulae apply

to it. However, instead of a constant head acting on the

16
IIt - Ze

turbine, we will have a variable head, which may become
greater or less than the normal, depending upon the change
of load. This variation in head tends to increase or
decrease the amount of power which would be developed for
the same gate opening at normal or constant head, and the
tendency of the fly-wheel effect to keep the speed constant
is lessened by the amount of increased or deficient power
so caused.

"d effective", as calculated in Chapter II, must, due.
to the causes above mentioned, be modified by the following

formulae:
we _(. ,(D H) 3
For closing gates, @™'=-doere ts 2 (16)

ad eff.
(a (D B)) 3 (17)
-—ET / 3

(For the derivation and explanation of these formulae

 

For opening gates, a” «

see Appendix C).

(D H)
H

interpolation from Tables IV to IX inclusive, or can be

can be obtained directly or by means of

computed from formula (15).
For illustration we will solve several problems:
(a) Given: L = 400 ft. H m 100 ft. N =»
2,000 H.P. 7 = 300 R.P.M. Wr mw» 400,000 ft” ibs,

Penstock Diameter: = 100 ft. of 5' 9" Dia. 3/8" plate steel
100 ft. " 6' 6" ™ 5/16" * "

100 ft. * yi oon 1T aM T 1?
100 f%. ™ 7' 0" " 3/16" " Mar
IilI - Se

Efficiency of Turbine: Full load, 2,000 H.P. = 83%
3/4 " 1,500 H.P. = 85%
+ " $1,000 H.P. mw 82%

4 n 500 H.P. x 70%
Discharge of Turbine: Full load, 2,000 H.P., Q = 210 ft® sec.
3/4" 1,500 H.P., Q = 156 ft° sec.
% " 1,000 H.P., Q = 108 ft? sec.
a oN 500 H.P., Q m 63 ft” sec.
Penstock velocity: Full load, 5' 9" dia., v. = 8.1 ft.per sec.
" om 6° 6" " woe 6.35" " 8
1 8  o9t on on oy. we 5.45" 7 8

Average v.= 3) 19.90 (6.63 ft.gec.

3/4 load, 5' 9" dia., v. = 6 ft. per sec.
v? w 6° 6" a | Vv. = 4.75 ? " wv
Lj r? 7 Oo" ve Vv. - 4.05 v w vv

Average v. = 3) 14.80 (4.93 ft.sec.

* load, 5' 9" dia., v. = 4.15 ft. per sec.
of 6" 6" Ve 2 35.00 " " "
nom o7tom os yee 2.60"7 7 8

Average v. = 3) 10.25 (3.41 ft.sec.

4 load, 5' 9" dia., v. = 2.42 ft. per sec.
vv a | 6' 6* | Vv. te 1.91 wv a | "T
¢ ve 7 oOo" WF Vv. = 1.63 vw ve wv

Average v. - 3) 5.96 (2 ft. sec.
18
III — 4.

L 400 4 2L _ 2x 400
H 100 a a

From table, a = (Average d = +" Average D = 6' 5")

2L 2x 400
a = 2.530 = 0.34 seconds.

If, instead of the tabular value for a, the general
value 3,500 had been used, then 24 = 0.242 seconds, which is _
somewhat smaller than the more exact value, but since Tf is
usually chosen several times larger than zh it is generally
satisfactory to use the general value of a. However, if by
using the general value of a the result from 2% should become
nearly 2 sec., it may be well to check this result with the
more exact value of a from the table or formula (13).

The regulating time fT must now be chosen, and the writer
would recommend that this be never taken less than 2 seconds,
unless with small turbines and then only if it be absolutely
necessary to get the desired regulation. It should in no
case be less than 1 second.

We will assume a regulating time T of 2 seconds for the
solution of our problen.

We now have complete data to compute the pressure
variations, from formula (15), or to take these values
directly from Table V which corresponds to our regulating

time Tf of 2 seconds. _L -_ 400 — 4.
H 100

19
III - 5.

 

For 100% load, Average v. 6.63 ft.sec. + {2 H) = 140.51 & 0.34

H

"75% " " ov. = 4.93 ft.sec. " = +0.36 &—0.263
" 50% " " v. = 3.41 ft.sec. " = +0.24 &—0.191
1 25% " " v. = 2.00 ft.sec. " = +0.132 &—0.116

Our next step is to find the effect of these pressure

variations on the speed regulation.

We proceed to find d@ eff. without any regard to pressure

variations, the same as in Chapter II.

d m= 800,000 x WH x Tf (3) w» 800,000 x 2,000 x 2 = 0.09

n® x Wr 300 x 300 x 400,000

Specific speed K = R:B-MY HP. , 300} 1,000 . 39
H YE 100/ Y760

*(For the speed of 300 R.P.M. this would be a twin turbine,

= 9%

 

 

therefore the power of one runner only must be used here).

From Table I, App. A, we see that the runners here used
correspond to standard runner Type C, and from Table II, App. A,
we find that the coefficient 0.8m = 0.69.

Therefore d eff. = 0.69 da = 0.69 x 0.09 = 0.0621 = 6.21%
We now proceed to find d eff. for partial loads: in this case
for 75%, 50% and 25% of total load. For diagram see Figure Le
Without regard to pressure variations, the speed regulation

would therefore be as follows: 100% Joad change, 6.21%

15% " 5.3 %
50% " fn 3.8 %
25% " " 2.1%

If we now substitute these values of 4 eff. and the values
+ {D 2) found on page 2 of this Chapter in formulae (16) and
(17) we obtain the actual speed variation, with the pressure

20
IIT - 6.

variations considered.

 

For closing gates: (Load thrown off) ad™. 4d eff. 14 PE 2 (16)

 

100% Losa: {2-H} » 0.51 ad eff. © 0.621 a? « 0.115 = 11.5%
75% * nm = 0.36 " = 0.053 " «£ 0.084 m= 8.4%
50% " n= 0.24 " # 0.038 " = 0.0525 = 5.25%
25% " " 2 0.132 =" » 0.021 " = 0.025 w 2.5%

d eff.
For opening gates: (Load thrown on) ad". OES (17)
H /2

100% Load: {p HD = 0.34 deff. = 0.621 a £ 0.116 = 11.6 %
75% " " = 0.263 " = 0.053 ™ © 0.0635 © 8.35%
50% " " 2 0.191 =" = 0.038 " = 0.0522 = 5.22%
25% " nm" m= 0.116 "™ © 0.021 " f& 0.0252 = 2.52%

It will be noted that in this case the speed variation for
load thrown on is practically identical with that for load
thrown off. This, however, is not always the case, rather the
exception, which may be readily seen from the tables for
pressure variations. The value for + 1D H) increases much
faster than that for — (DA) and may become infinity, whereas
the latter can never exceed 100%.

For a load change of 100% the speed variation is almost
doubled, when the pressure variation is taken into consideration,
whereas for a load change of 25% it is hardly affected. Since
a load change of 25% is very seldom exceeded in most power
plants, the effect of the pressure variations on the speed
regulation is not as bad as would appear at first glance.

21
iL

 

—
om
t®

—_—
s —

—!

~_

1

—w
. ae

 

’
—
. ~
°
—
e _
s
Iil - 7.

As we have already stated, it is uncommercial to
provide against something that may never happen, and if the
greatest load change ever expected under normal operation
is 25% it would be a waste of money to provide for 100%
change of load, as far as speed regulation is concerned.

From observation of the tables on pressure variations
it will be noted that a change in velocity of one foot
will have considerable effect on the pressure variation,
and this velocity should always be kept as low as is con-
sistent with a reasonable investment. We should always
keep in mind that by decreasing our penstock velocities
we not only help our pressure variations, and hence speed
regulation, but improve the economical use of the water,

as our friction head will be less the smaller the velocity.

22
 
CHAPTER IV

PRESSURE VARIATIONS AND SPEED REGULATION
AS AFFECTED BY THE PRESSURE REGULATOR.

If in the problem in the foregoing chapter the
pressure rise of 51% for 100% change of load should be
considered dangerous to the penstock, on account of the
stresses on the material which would be imposed, we can
overcome this danger in one of three ways:

(a) By decreasing the velocity in the penstock, thus
decreasing the pressure rise to a point which may be
considered safe.

(b) By increasing the factor of safety in the design
of the penstock.

(coc) By introducing a pressure regulator of sufficient
size to reduce the pressure rise to a safe point.

(As stated in Chapters I and III, depending upon the
load it may not be necessary to consider a greater load
change than 25 or 50% from the standpoint of speed regula-
tion, yet it is always necessary to consider the effect of
the maximum pressure rise for 100% load change, since in
the case of a short circuit or break in the governor such
an extreme load change may occur and subject the penstock
to the corresponding pressure rise. )

The object of this chapter is to discuss the remedy
indicated under (c) and to show its effect upon the speed

regulation.
25
From tests made by the writer and substantiated by others,
the pressure rise in a turbine, and consequently in the penstock
feeding a turbine, depends directly upon the discharge capacity
of the pressure regulator, if properly designed, and upon the
sensitiveness of the operating mechanism of the pressure
regulator. The latter should be adjustable, so that the
pressure regulator can be set to meet the requirements of any
fixed conditions after being installed.

If the sensitiveness of the pressure regulator is 10%,
that is, if the operating mechanism is so adjusted that it will
open if a sudden load change amounting to 10% of the total load
occurs, the pressure rise will be as follows:

Discharge capacity, 100% of turbine discharge, Pressure rise 10%

" " 75%" r " i " 20%

i " 50%" " m " n 30%

Due to the insensitiveness of the operating mechanism the
pressure rise will be the same regardless of the load, and will
be even slightly greater for a small load change than a large
one, as it will take proportionally more time to open sufficient-
ly to pass the small quantity of water. Another factor which
helps to bring this about is the discharge coefficient of the
pressure regulator, as this is greater the larger the opening.
In calculating the effect of the pressure rise on the speed
regulation it is therefore necessary to take this into consider-
ation for a partial load change as well as a full load change.

As explained in App. C., although — 2) is always less

than + {DEI » the corresponding regulation as affected by these
24
 
pressure variations is practically the same, at least within
the usual limits. If this is the case, it would appear that
&® pressure regulator is of no value to improve speed regulation.
However, in most plants the load is always thrown off much more
suddenly than thrown on. This is true in most plants, with
few exceptions, such exceptions usually occurring in small
plants, and they always require more careful consideration
with respect to the subject of speed regulation.

A lighting load is usually turned on as suddenly as off,
but its proportion to the total load is exceedingly small, a
few lights being usually turned on at a time. Motor loads
are usually turned on very slowly by meens of resistances, and
such sudden loads as may come on the motor during its opera-
tion are generally only a small part of the total load. On.
the other hand, a motor load is usually thrown off suddenly,
whether purposely or accidentally, so that it becomes more
difficult to maintain constant speed for load thrown off than
for load throvm on, and it is of great benefit to have con-
ditions for good regulation better for that case.

This benefit is obtained by means of the pressure
regulator, as + PE) nay be reduced to such an extent that
its effect on the speed regulation is very small.

It is of course evident that a pressure drop less than
100% is not dangerous to the penstock, whereas a pressure
rise much less than 100% may be very dangerous, particularly

if it occurs frequently, in which case it is liable to

£5
IV = 4,

erystallize the penstock material, especially at the rivetted
joints, so that the penstock may give way, with a pressure
rise much within the limit set by the original factor of
safety. In such @ case the pressure regulator not only
assists in the regulation of speed, but protects the penstock
as well.

When a pipe-line or penstock is laid over a rough country,
or where it descends down precipitous hillsides after running
along a comparatively gentle slope, care must be taken that
the water will never be accelerated so quickly at a change in
slope that the water column could break and thus exceed or
even approach the maximum pressure drop of 100%. In such a
case the penstock is in danger of either collapsing or of
being shattered by a possible water ram.

For illustration we will consider the following problem:
L=1,800 ft. Hem 300 ft. N=7,500 H.P. n = 400 R.P.M.

Wr” = 600,000 ft” lbs. = 265 f° sec.

Q max.
Average maximum penstock velocity m= 7 ft. per sec.

= SO. = 6 zh = 2 = 1.09 seconds.

Make T — 2.5 seconds.

q = 800,000 x N x T _ 800,000 x 7,500 x 2.5
n° x Wr“ 400 x 400 x 600,000

 

= 0.156 = 15.6%

Specific speed, k = az} = =«: 27.8 — App. Type C. single

runner.

From Table II: 0O.8x = 0O.69 .
d eff. = 0.69 x 0.156 = 0.108 = 10.8%

26
From the diagram, Figure 3, we have:

For 100% load change = 7,500 H.P. Speed variation = 10.8%

n 95% © " = 5,625 H.P. " " = 9.1%
" 560% " " = 3,750 H.P. " " = 6.4%
n 695% +" " = 1,875 H.P. " n = 3.25%

From Tables V and VI, by interpolation we have:

{2H = 0.70 and max.— O2t = 0.445 = 70% and 44.5%

The pressure drop of 44.5% is not dangerous to the

max. +

penstock in any case, but the pressure rise may be,
especially if it did occur frequently. If we assume this
to be so in this case, we know from what has gone before
that by means of a pressure regulator with a capacity 75% of
that of the turbine we can reduce the pressure rise to 20%
for all loads.

From this we will get the following a™:

 

 

Load Change : Pressure : Penstock : Speed
: Variation : Velocity >: Variation
; + 20 % : : + 14.2%
7,500 H.P. : — 44.5% : 7) 6ft. > = 26
- 5 + 20% : + 12%
5,625 H.P. : — 31.7% > 5 ft. > — 19.1%
+ 20% : + 8.4%
3,750 H.P. : — 23.3% > 3.5 ft. >: —- 16. %
: bt 20% OO | : + 4.27%
1,875 H.P. : — 14.3% > 2 ft. > = 13.5%

 

For most plants , where several units of this size are
installed, the writer would consider this satisfactory
regulation. A load change exceeding 2,000 H.P. will seldom

27
IV - 6.

occur, and probably never on one unit, and only for
load thrown off; for load thrown on it is doubtful
whether 2 to 3 % speed variation will ever be
exceeded.

However, if it should prove desirable to improve
the regulation slightly, the penstock diameter should
be increased, unless it is found cheaper to add

additional fly-wheel effect.

28
V-1;

CHAPTER V.

PRESSURE VARIATIONS AND SPEED REGULATION
AS AFFECTED BY THE EQUALIZING RESERVOIR OR STAND-PIPE.

The stand-pipe should be located as near the turbine as
possible, @as has been previously stated. If it is arranged
with an overflow, the pressure rises can be practically
eliminated, and the pressure drop will devend directly
upon the size or capacity.

The minimum height of such a stand-pipe is determined
by the restriction that in no case must the water level
in it drop to such a point as would admit air into the
penstock. This condition is satisfied by the following

 

equations: (See Figure 4.)
a =} 9 Lxv (18)
€
and
a =z D+Hf (19)

Where Q — cu. ft. per sec. F == area of stand-pipe in
sqe ft. UL «= pipe line length above stand-pipe in ft.
Vv = pipe-line velocity in ft. per sec. H f = friction
head in ft. D = diameter of penstock in ft.

It is seldom satisfactory to have the stand-pipe
overflow, and furthermore it is uneconomical, so that
stand-pipes are usually built high enough so that the
water, even with a maximum load change, cannot overflow.

29
In northern climates, where there is danger of freezing, the
entire stand-pipe should be well lagged, and quite often it
must be provided with steam-pipe coils, supplied with steam
from a boiler installed at the foot of the stand-pipe.

In order to reduce the height of the stand-pipe as much
as possible, the slope of the pipe line should be as little
as is consistent with the velocity head and friction head
required, as the top of the stand-pipe must in any case be
higher than the level of the water in the fore-bay, to meet
the conditions of a shut-down with pipe-line full.

The actions of the water in a plant with stand-pipe are
easily explained, but very difficult to compute. It will be
assumed that the stand-pipe is located less than half the
total length of the pipe-line away from the turbine, and
that the slope of the last half or less of the pipe exceeds
the rest considerably, as is always the case in practical
plants.

If load is suddenly thrown on, a certain time will
elapse before the water accelerates in that part of the
pipe between the stand-pipe and the turbine, another and
longer time elapses before the water in that part of the
pipe above the stand-pipe accelerates to the velocity
required by the new load. During the latter time the
stand-pipe must supply the turbine with a quantity of water
which is the difference between that used by the turbine
and that supplied by that part of the pipe above the

30
 
standpipe at the reduced velocity during acceleration. The
reverse is true when load is thrown off.

The time # required for accelerating or retarding the
flow of water from the velocity Vo to Vy» oF the time
required for return to normal head after a change of load

is found by the following equation:

FL (20}

Ag
Where:
= cross sectional area of the standpipe in sq. ft.
™ length of pipe line above standpipe.
= cross sectional area of penstock in feet.

= acc. of gravity 32.2 ft. sec.”
= 3.1416

Q 7m 3. H
i

The drop of the water level below the hydraulic
gradient in the standpipe can be found from the following

equation:

po _2HD= - 54 [Z (v2 ¥0%) + BF (vy - v9] (21)

where:

H = gross or static head on the turbine.

A, F and L, the same as in equation (20)

Vo m the velocity in the penstock at the instant of load
change.

V1 = the velocity in the penstock required for the new
load.

And the total drop below the fore-bay level will then

31
2g

be: Dp = D+0C Vo (22)

where o Vor

=/ the friction head in the pipe line above the
standpipe, where:

GC = the friction coefficient for flow in pipe lines =

1 "
Hs +f¥ + ote.) (23)

f for the usual plate steel penstocks is app. 0.015.
(It will be noted that the equation (21) is a quadratic
equation. )

The upward surge can be found by the same equation, by
& proper change of signs, but is unimportant since it is
always less than the dovmward surge, and the maximum height
of the standpipe must be determined from the surge or rise
obtained when the whole plant is suddenly thrown off, as in
the case of a short circuit. This maximum rise of the
water in the standpipe above the fore-bay level, when full

load is thrown off, can be computed from the following

equation:

2. Ay e/k _ef
Da~ = Yo ( 3 0) (24)

in which # is the value from equation (20) and ¢ that from
equation (23); all other values are the same as in equation
(21).

(For the derivation of these formulae refer to "Water
Power Engineering" by D. W. Mead, page 696. )

All of these equations hold equally well for calculations

with respect to a reservoir or a standpipe.

52
 

 
Equalizing reservoirs are usually provided with a
spillway so that their maximum height need not exceed that
of the fore-bay level. In order to reduce the maximum
height of the standpipe, some plants are provided with a
water rheostat, which can load the turbines from the low
tension buss-bars in case of a short circuit or other sudden
large load change. Such a rheostat must be thrown in by
an attendant in the plant, but the writer believes the
standpipe should be built as high as necessary, disregarding
such safety devices. Such rheostats are also often used
to assist speed regulation, and may prove of great value in
connecting two or more units in parallel.

To make the given formulas clear we will solve a
typical problem, as follows:

Assume a plant of four 8,000 H.P. units operating under
ea head of 350 ft., and that a load drop of one-third the
total is to be provided for. There are four pipe lines,
each 6.75 ft. in diameter and 4,000 ft. long, connected to
a single standpipe by means of a header. The distance
from the standpipe to the turbine is 600 ft.

At two-thirds load each pipe discharges 155 ft° sec.
with a velocity of 4.5 ft. per sec. At full load the
discharge is 245 £t> sec., and the velocity 7 ft. per sec.
Therefore Vv) = 4.5 and Vv, = 7.

We will assume a standpipe 30 ft. in diameter.

55
Then A=4x35 = 140 sq. ft.

¥ — 707 Sde ft.
L — 4,000 ft.

yi x 4,000
= P L — e 2 — e e
tT =a] > 5.14. 140 x 32.2 78.5 seconds

Then from equation (21)

 

 

 

2 _ 2A IL 2 2 H f

D ~2np=-2A [3 (2 vo ) + EE (x — ¥4)|

2 _ _ 2x 140 [4,000 (2 _ 4 52) 4 350 x 78.57 — 5)
p? 2x 350 D=— 2% 140 [ sa2e2 4.5%) + 350 x 78.5(7 — 4,5)
pD* — 700 D = — 9370.

D” — 700 D + 9370 = 0
2 2
4D” — 2,800 D — %00 = %0 — 4 x 9370
=
2-700 = +\ aq0'- 4 x 9,370

D = 700 — {700% 4x 9,370 = 700-673 _ 15.6 +.
2 2

From equation (22)

 

 

 

and equation (23)

1 L 1 4000
= 2 ( +53) ea sone) =
2
D, = 13.5 70.157 x 75 = 16.27 ft.

This value does not appear excessive, and if the standpipe
is built in the shape of a water tower, as it should be for our
case, an allowance of 20 ft. below fore-bay level should be
sufficient.

We can now find the maximum height necessary from
equation (24)

34
2 A_e/(L_st
Da = § Vo € 3 r)
Vy in this case is the full load velocity = 7 ft.

All other values are the same as before.

Dee ns 240 x 7 x 7 (4,000 _ 0.187 x 78.5 x 1)
707 $2.2 mo)
Da = 31 ft.

The total height of the 30 ft. tank of the stand-pipe
would therefore be over 50 ft. and should be constructed
approximately as shown in Figure 5.

By increasing the diameter, the height can of course
be reduced as may be desirable.

We will now investigate the speed regulation both with
and without the stand-pipe, under the assumption that there
will never be less than two wnits running, and that the
largest sudden load change will not exceed one-third of the
total of 32,000 H. P.

(a) Without stand-pipe:

 

 

 

H 350
Maximum load change = 11,000 H. P. on two units of
8,000 H. P. each. n — 327.5 R.P.M.
z 2 2u 2x 4,600 _
Make T = 5 sec. v= 7 ft. per sec.
~ 800,000 x N x f 800,000 x 8,000 x 5
ad = 2 == —2. 2 “> = 0.42 42
~+2H _ 0.757 and —~DE _ 0.426
H '

From these results it can be seen at once that the regulation

35
would be far from satisfactory. If we reduce the regulating

time to 4 sec. we decrease ad, but since DE wilt thereby be

H
increased and d™ will be no better than before.

(bd) With stand-pipe:

L = 600 ft. LTppp, = 600 + 300 (length of stand-pipe

L 900 2L 2x 900

a= 350 = 2-5 a = "3,300 ~ 0:548 seconds.

v—-7-—-4.5 =2.5 = DV to be used in formula (15)
or to get +DH from the tables.

For @& regulating time of 3 seconds the pressure
variations, both plus and minus, do not exceed 6%, so that
their effect upon regulation may be disregarded.

The maximum pressure rise, if the total load of

32,000 H.P. is thrown off will be 20%.

800,000 x 8,000 x 3
327.5 x 327.5 x 700,000

R.P.M./ H.P. 327.5 ¥ 8,000
Specific speed kK=- ——— = = 19.4
P P i Vi 350 | 350

Approximately Type B runner and 0.8 = 0.703
dopp = 0.255 x 0.703 = 0.18 = 18%

= 0-255 .— 25%

 

From the diagram, Figure 6, we see that the maximum speed
variation will be 13.75% for a load change of one-third the
total, if distributed over two units. This would undoubt-
edly be considered satisfactory for most cases.

Although not so economical in the use of water and

not at all required to reduce the pressure rise, it will

356
 

 
often prove cheaper to install a pressure regulator rather
than build the stand-pipe to the height as found necessary
by equation (24). With a suitable pressure regulator in
the system, the height of the stand-pipe above the fore-bay
water level is more or less a question of judgment, but if
made equal to the depth it will always be sufficient. It
should be remembered, however, that & pressure regulator is
only a mechanical device, and somewhat complicated at that,
and many engineers would prohably be unwilling to take the
risk of a catastrophe in case of a sudden full load change

with the pressure regulator out of order.

37
VI -1

CHAPTER VI.

PRESSURE VARIATIONS AND SPEED REGULATION
AS AFFECTED BY A COMBINATION OF STAND-PIPE AND PRESSURE REGULATOR.

The capacity of a stand-pipe must be the same regardless
of the distance it is located from the turbine, provided the
other conditions, such as maximum change of load to be pro-
vided for and regulating time are the sane. If a greater
regulating time is allowed on account of having a longer
penstock, the capacity of the stand-pipe could be theoretical-
ly reduced slightly, but this should not be done in practice,
and the formulae from the foregoing chapter therefore hold
good for any conditions.

If in the problem in the preceding chapter we assume a
penstock 1,800 ft. long in place of 600 ft, with every other
condition remaining the same, our final results as to speed
regulation will be altered, as is immediately evident, since
# becomes considerably greater, and therefore the effect of
the pressure variations, which we neglected in the first
case, must now be considered.

The speed variation of 13.75% for the maximum load change
of 11,000 H.P. on two units should not be exceeded, as this
is already somewhat high. Assuming that the penstock
diameter can not be increased, due to engineering limitations
of some kind, it becomes necessary to either determine upon
a shorter regulating time or more fly-wheel effect in order

to keep the speed variation at least within the first figure

of 13.75%.
38
 
VI -@2

 

a 2 2(1,800 + 300) _
Our new critical time = = 3,300 = 1.27

seconds. It would therefore be impossible to choose a
shorter regulating time than 2.5 seconds as against 3 seconds

in the previous case.

— 327.5 x 327.5 x 700,000

and deff. = 0.213 x 0.703 = 0.15 = 15% (See Figure 7).

 

For a load change of 11,000 H.P. on two units the

velocity change will be as before, namely 2.5 ft. per sec.

 

L — 2(1,800 + 300) _
z= 350 = 6 and, from the tables,
(D H)
+ “7 = 0.70 = 70% for a total load change, and if we

provide a pressure regulator equal to the capacity of the

turbine this will be reduced to 10%.

 

 

 

— iD (for a load change of 5,500 H.P.) = 0.175 = 17.5%
dort. .
and a" ont g = O_o. 75 = 17.5%

= (2 . (DE) 3 (1 -0.175) 3

This result is not quite as good as before, even with
the reduced regulating time. We could use a regulating
time of 2 seconds but, if at all possible, the writer would
recommend a larger fly-wheel effect, either in the shape of
a separate fly-wheel or additional weight in the generator
rotor.

For load thrown on,the regulation is of course better
and, with a maximum pressure rise of 10% and load change of

5,500 H.P. as before, amounts to 12.4%.

39
VI - 3

By means of the pressure regulator we therefore
improve our regulation, protect our turbine and penstock
and, as stated in Chapter V, reduce the height of our
stand-pipe.

40
VII- 1

CHAPTER VII.

THE EFFECT OF THE SYNCHRONOUS BY-PASS AND THE DEFLECTINUG
NOZZLE UPON SPEED REGULATION.

In a plant with long penstocks, where it is impossible
to install a stand-pipe, and where it is practically out of
the question to increase the size of the penstocks on account
of excessive cost or otherwise, it will often be found
impracticable to provide sufficient fly-wheel effect to
obtain satisfactory regulation. In such a case &
synchronous by-pass must be provided in the case of

reaction turbines. Such a synchronous by-pass should be
exactly what the name implies, that is, it should discharge

that part of the full load flow of the water which is not
passing through the turbine at any given time. If the flow
of the water through the turbine is changed in volume, the
change in flow through the by-pass should correspond.

It is very difficult to design a by-pass that will do
this, since the discharge through the turbine gates is not
proportional to the governor stroke, from which the
by-pass must be operated, and furthermore the coefficient
of discharge of the by-pass can not be kept constant during
its full range of opening. Due to these causes and the
insensitiveness of the governor, the pressure variations
both ways may be quite considerable with the synchronous
by-pass and should be considered in computing the speed
regulation. They may be as high as 25%, but the writer's

41
VII - 2

experience shows that they can be kept as low as 10% for
both opening and closing gates. The by-pass should
always be brought as near as possible to the turbine gates
and in line with the flow of water into them.

The problem therefore becomes identical with that of
Chapter III and the illustration there given covers the
case, with the exception that the pressure variations must
be assumed instead of taken from the tables on pressure
variations. In the case of impulse turbines, the
synchronous by-pass may also be used, but the deflecting
nozzle is more generally used in practice, as it has
several important advantages. With the latter the flow
of water is not interfered with, but the stream is partially
or wholly deflected from the buckets, maintaining practically
constant pressure, and varying the active volume only.

The problem therefore reduces itself into one identical
with that of a turbine in open flume, or without pressure
variations, and is covered by Chapter II, where the

regulation is subject only to the fly-wheel effect.

42
A - (a)

 

APPENDIX A
_ O J T
DERIVATION OF FPLY-WHEEL FORMULA da = B00 oo xz pa x Wr
M ve
The energy of a rotating mass is P

If the peripheral velocity of the mass, M, is reduced

from Vo to v, it will lose energy amounting to: M (7, —_ v7
3 so

The factor (v5 v,”) can be transformed algebraically
as follows:

(o®m17)= (vot va) (rem)

Substituting in the above, and multiplying by + we have:

Energy lost = Mv (v2 +) (Vez — v3)
¥ 2 Vv
but vo + V) _ the average velocity = V
~».~—= =

and Vp — — the reletive change of velocity =a (dis
“2=5 Vl = ng

also called the per cent of ununiformity. )

Substituting the values V and d in the above
equation of energy lost, we have:

Energy lost =Mvva=-=-M va, or

Mu (v2_ v2) Mv? a

2

43
oa
A - (bd)

Referring to diagram, Figure 8, let ordinates
represent H. P. and absoissas represent time.

Then the energy it is that which must represent
the amount given out by the rotating masses, after the
turbine gates have been closed off, and T is the time
required to give it up.

That is, if the total load is taken off suddenly,
the power would drop from a to b, but on account of the
rotating masses this would not happen instantaneously,
but would take a certain time f, and the effect of the

masses is equal to the area below the line y, and if this

NT e
2

Then a 550 (ft. lbs.) = M v® a.

is a straight line the area —

Or

NP 560 _ WoW r® n’ 4x4
2 g 60 x 60

Where M = Wand v = 27 ot m

44
A - (¢)

Substituting numericel values and transposing, we have:

wre — 3 800,000 (1)

T 60x
n*d 2x

0

or? n® a (2)
“WH x 800,000

also
_ 800 900 x Yut (3) which is the formula as
first given, and
Wr® = Moment of inertia of the rotating
masses in lbs. ft. squared.
T $= Time to close or open gates by the
governor in seconds.
n <=R.P.M.

= Ve _ Vy = Mo —%1 = the relative
v n

_ change of speed, = (DH)
— Tn Angular velocity.
w = "xo- = gular velocity

(Dw)= Change in angular velocity.

Correction: In deriving the above formula it was
assumed that the line y wes e straight
line and the area under it was equal to
e under the assumption that the total load

was thrown off.

45
A - (a)

However, the friction load of the turbine and
' generator remains, and the actual area represented
will be that under a curve f (y). The effect of
this friction load will of course be different depend-
ing upon the various types of turbines used, but the
original formula may be modified safely as follows, the
curve f (y) being approximately a parabola.

a' — 0.8 a (4)

Another cause which should be taken into consideration,
providing the turbine is properly designed so as to give
its maximum efficiency at normal or synchronous speed, is
that the efficiency is reduced with either inoreasing or
decreasing speed, which has the tendency to still further
reduce the area under the curve f (y). The amount reduced
varies with the type of runner, being greatest for e high
speed, high power runner, and least for a low speed, low
power runner, since, if correctly designed and therefore
following the laws of hydraulics, the change inefficiency
will be greater for a high speed than for a low speed

runner for the same per cent of variation in speed.

46
A - (e)

The formula modifying dgrr for this cause, is as follows:

a'
der? = 7 a (5)

4-
ny) _y
No

 

 

Where "1 varies from 1.8 for a runner of low speed

vo to 1.3 for a runner of high speed.
(Referring to the table of standard runners types A to F,
designed by the writer for the Allis Chalmers Co., the

values of ™l are as follows:

 

 

 

 

 

 

 

 

 

 

 

 

No
TABLE I.
Type of Runner A 5B | ¢ Die |F
Value of _"] 1.8] 1.7] 1.6] 1.5) 1.4] 1.2
No
Pa
Specific Speed k =RE UV. 13.55) 20.3| 29.4/40.7] 62.8]83. 76

 

The specific speed is that speed in R.P.w. of a turbine
runner, €iminished in all dimensions to such an extent as to
develop 1 H.P. when working under the head H=1 ft. (This
value is called "Type Characteristic" by Prof. Zowski in his
article in the Michigan Technic of June, 1908. "The american
High Speed Runners for Water Turbines.")

To simplify calculations, the following table will prove
efficient:

47
-
A - (f)

TABLE II.

 

Type of Runner A B C D E F

 

0.8% in dgpp = 0.8md | 0.714/0.703/0.69/0.671] 0.645 |0. 606

 

 

 

 

 

 

 

 

 

For impulse turbines the same dgrr as for type
A reaction runner should be used.

To find the speed variation for any other than a full
load change, the following must be considered.

If T is the regulating time for the total governor stroke
or 100% load of the turbine, it would appear that for 50%
load or stroke, the regulating time would bes, and for 25%
> etc. As far as the stroke is concerned, this is nearly
true with a mechanicel governor, and the speed variatiors for

part load chenges can be calculated as follows:

The formula for d which we have given mey be transformed

 

 

 

to appear:
2(mz£) | 2(@e }
Z ey 3 Xx 2
550 hN 550 NT

In which the new symbols 4 end a are respectively, the

change in loed, and the remaining loed.

48
A - (g)

 

 

 

 

1
If we let S represent ela/m ¥ 7 in the foregoing formula,
it will appear as Cy |
(550 NT
follows:
a=sz? (1s 424) (7)

From this formula we can now compute d for various pert
loads, since S, as may be readily seen, is ad as figured from
the original formula for total change of load or S$ = d' ert
in the ebove formule, and for load changes of 25%, 50% and
75% we have the following:

(a) Por 25%, Z = 0.25 and a = 0-76

4'2570 = dere x 0.625 ( 1-@ore x 0.25)
(bd) For 50%, Z — 0.50 and a — 0.50

4'50% = d'opp x 0.25 ( 1 — a' ore x 0.333)
(co) Por 75, Z = 0.75 and a — 0.25

a'75% = d'orr x 0.56 ( 1 — d' opp x 0.25)

The results of these computations plotted appear as

in the following diagram, Figure 9, line marked a.

49
A - (h)

Line a correctly represents the speed
variation under the assumption that the governor acts
instantaneously upon a speed change; however, a certain
speed change must occur before the governor can act on
account of the insensitiveness of the fly-balls, and
the actual speed changes are more correctly represented
by a line _b , midway between the curve a anda
streight line o as shown. ©

For a hydraulic governor, an entirely different
line of reasoning must be followed. My own observe-
tions are supported by those of others, who claim
that the regulating time for all gate openings is neerly
constant and equal to that of total gate opening for a
hydraulic governor.

This is due largely to the throttling effect of
the regulating valve and the time required to overcome
the inertia of the regulating masses. That this fact
is not detrimental to the speed regulation can be seen
from what follows, and that it is of large benefit to
the problem of pressure variations, and therefore
indirectly again to regulation, will be seen from a

study of the chapter on pressure variations.

50
-—
A - (i)

From. experience and tests it has been found that
for hydraulic governors the curve a, shown under
partial load speed variation for mechanical governors,
must be modified as shown in the following diagram,
Figure 10.

The curve d is an are of a circle drawn tangent
to the line 0 — dad, which is the line of apparent
regulation, and through the point dgrpr, which is the
actual value of d' as modified for various causes as
given.

It would apvpear from a comparison of the regulation
curves given that the speed variation would be less for
partial loads, with the mechanical governor than with
the hydraulic governor. This, however, is not true
for practical purposes, since the regulating time for
any but very small mechanical governors is of necessity
very great compared with the hydreulic governor, the
regulating time of which can practically be made as short
as desired for any size. From an investigation of the
original formula for 4 it can be seen thet 4 varies
directly with T and therefore if we cen use a shorter

time T with the hydraulic governor than with the
mechanical, it is seen that the advantage which is

apparent can be overcome many times, due to the very

short time T possible.

51
A - (3)

A mechanical governor can only perform a definite
amount of work per unit of time, which depends upon
the size or strength of the driving belt or gears
which must transmit the power of the governor.
If, for instance, a mechanical governor is capable,
due to the strength of its parts, to transmit 1H.P. —
550 ft. lbs. per seoc., and the work required to regulate
a turbine is 11,000 ft. lbs-, then the regulating time
must be 11.000 — £0 seconds. |

The work which a hydraulic governor is capable of
doing is a question of the size of the regulating
cylinder, and the time in which it can do this work
depends upon the time in which sufficient oil can be
brought to the cylinder. This time again depends upon
the pressure in the pressure tank, the size and discharge

coefficient of the oil pipes and the reguleting valve.

52
rm cy
.
B - (a)

APPENDIX B

FLY-WHEEL EFFECT. Wr

Fly-wheel effect is the capacity of a rotating mass
to store or provide energy. This effect is expressed
in lbs. at 1 ft. radius and is equal to the moment of
inertia of a mess, that is, the weight by the square of the
radius of gyration, or lbs. ft.® Modern alternating
current generator rotors are built in the shape of a
common fly-wheel and usually have sufficient fly-wheel effect
to obtain satisfactory regulation. The fly-wheel effect
of the rotating parts of the turbine are usually not con-
sidered, but their value is variable for different types
of turbines, and should be carefully considered for large
low head multiplex and large dia. impulse turbines.

For smaller turbines, their value may be regarded
as a factor of safety. For very large dia. and high
power reaction turbines, the weight of the water in the
runner must also be considered as its effect is exactly
the same as any other rotating mass, and its consideration
is necessary to make the calculations of any value, as the
proportionate effect of the water may be a large per cent

of the total rotating mass.

53
—_—

Se ee ee
B - (b)

With geared turbines the effect of all geers
and shafting should be considered.

Where turbine driven direct current exciters
are used in a large hydro-electric plant, their
possible regulation should be carefully considered,
as the fly-wheel effect of these exciter generators
is usually small and an additional separate fly-wheel
may prove necessary. .

When purchasing generators, a comparison of the
fly-wheel effect should be msde, as they are usually
built as light and compact as possible. Additional
fly-wheel effect within certain limits can usually
be obtained at a very small cost in alternating current
generator rotors, end will usually prove cheaper and
more convenient than to add a separate fly-wheel, in
cases where the original rotor effect is not
sufficient.

If a separate fly-wheel must be added, its weight
for a given moment of inertia or fly-wheel effect
depends upon the permissible peripheral speed. In
calculating a fly-wheel the runeway speed (speed for
full gate opening and friction load), must be con-

sidered and is really the determining factor. For

54
B - (c)

an impulse turbine this runawey speed will be about
90% above normel. For the standard types of reaction
runners, 4, B, C, D, E and F, the characteristics of
which are already given, the runawey speed will be
respectively as follows: 70, 65, 60, 55, 50 and 45 %
above normal. These are the maximum speeds, and they
may be appreciably less, denending upon the setting of
the turbine ana “se type of bearings used. Wor low
speeds, fly-wheels of the ersine type may be used, but
for high speeds they shculd be of the disk type, turned
all over and carefully bealenced. The maximum peri-
pheral speed for solid cast iron disk wheels should not
exceed 100 ft. per sec. and for cast steel wheels of the
seme type it should not exceed 200 ft. per sec.

Cast steel wheels are usually more convenient
on account of their smaller size for the seme fly-wheel
effect, and cheaper because less additional bearing
surface is required, end the greater cost per ib. of
material, if any, is off-set by the lighter wt., the
fly-wheel effect varying with the equere of the radius
of gyration.

55
C - (a)

APPENDIX C.
DERIVATION OF PRESSURE VARIATION FORMULA.

We know,from the fundamental axioms of mathematical
physics, that a given mass acted upon by a variable force
in an interval of time is accelerated or decelerated.
This relation is given by the following fundamental
equation:

May = P and Mdv = P dt.

In the problem relating to speed regulation we always
deal with a fixed time T, and a variation in the velocity
of the water brought about in the time r, which is identical

with a pressure variation. We may therefore write

M(Dv) = (DP)t.

Where M = mass = x
° W = weight in lbs.
& = 32.2 =m acceleration of gravity
P = pressure —= Hy <= (D H)

where y = specific wt.
Let L represent the length of a pipe line or penstock
and F its area.

Then M = a
P=FP=Fy (D#)

56
 
c - (b)

And substituting in the equation MDv = DP t
we have “2 Z(p v) = Fy (DH)?
Transposing (DH) as a (9) = the actual increase
or decrease of head for a certain regulating time fF and a
variation of the velocity of the water of (D v).

This formula, to express the result in per cent. of
the total head, will read

(DH) _ L (Dv)

3 oe: (10) and

for both increase and decrease it will read

(DE) ~+ Lv) (41) where:
H Hel |
+ is for closing,

— is for opening gates.
From this formula we learn that the pressure variations
vary directly with the length of the penstock and the
velocity of the water in it, and inversely as the head
and the time during which the original velocity is varied
by (Dv).

If we assume an allowable pressure variation,

(D H)
E

 

the regulating time can be found from a transposition

57
C - (ce)

of the above formula, as follows:

 

+ L(Dv)
H g PH

&8s an example:
Let ZL = 500 ft.
H = 100 ft.
10 ft. per sec. for T =0

4
°
l

vy. = «iOft, " " " T=2 sec.

(Dv) = vo —vy = 5 ft. per sec.

 

 

which shows that for a change in penstock velocity of 5 ft.
per sec. the pressure would increase or decrease 39%.

Where a pressure variation not to exceed 10% is
(D H)
H

 

permissible, = 0.10 , the corresponding repulating

time must be for the same conditions:

+ 500 x 5

T= — 00 x 32.2 x 0-10

 

=+7.8 sec.

which shows that it takes 7.8 seconds to accelerate or
decelerate the velocity by 5 ft. in order not to increase
or decrease the pressure to exceed 10%.

As may be readily noted, the above formula gives

equal results for increase or decrease of pressure, and

58
C - (a)

that both values become infinity, if any of the factors

in it become infinity or zero. It is impossible, however,
to imagine the pressure decrease to become infinity: it is
evident thet the decrease can never exceed 100%, and if
this limit is surpassed, or even reached, the liquid
colum must sustain a rupture, and the water would flow
intermittently or in the form of pulsations. From this
it is evident that the formula can only be correct, if

at all, within certain limits and under certain restric-
tions, and then only approximately, as we will see from
our further discussion.

The above indicates that we must analyze this problem
further, in order to get formulae which will cover the
ground and give more exact results.

It is a well known fact that if penstock conditions
are disturbed by moving a gate anywhere in the line,
either at the upper or lower end or intermediately, the
whole system becomes oscillatory, which means that the

pressure and velocity in the penstock are oscillating.

A similar phenomenon can also be experienced in an
open flume in which water is flowing. Assume that the
gate at the lower end of such a flume is closed. A wave
will be produced next to the gate, and this will proceed
upward along the flume with a certain velocity, until it
disappears either in the intake basin, where v nearly

equals zero, or it will be dissipated by friction on the
59
Cc - (e)

flume walls, depending upon the length and roughness of the
flume and the velocity of the wave.

In closed flumes or penstocks such waves do not
disappear so quickly, the surface friction being small in
comparison with the velocity and the produced force, and
its influence may even be neglected.

It is evident that these waves, or rather vibrations,
ere liable to interfere with each other and consequently
affect the pressure in the penstock. It is therefore of
greatest importance to determine the velocity with which
these vibrations travel in and along the penstock.

This velocity of the vibrations, which we will
indicate by C in our further discussion, depends (a) upon
the compressibility of the liquid and (b) upon the nature
of the material of which the penstock consists.

It is a well known fact that vibrations in water
proceed with the same velocity as sound does through the
same medium, and, if the walls of the flume or penstock
can be considered absolutely rigid, (C therefore only
depending upon the compressibility of the water), the
velocity of the vibrations will be 4,650 £t. per sec. or,
as already stated, the same as that of sound. The
penstock walls, however, are always flexible to a certain
degree, depending upon the elasticity of the material
from which they are constructed, and therefore they also

exert a certain influence upon the vibrations.

60
c - (f)

Under the influence of pressure variations the
penstock walls, due to their property of elasticity,
expend and contract in a rather remarkable degree, called
"breathing" of penstocks. Due to this breathing the
velocity of vibration is reduced, the motion so produced
naturally having a damping effect upon the vibrations,
the same as any other obstruction would have. The
velocity of 4,650 ft. per sec. may therefore be considered
a maximum with which any vibrations in the penstock will
proceed.

The actual velocity of the vibrations can be computed

by means of the following formula:

&

ay
i,iD (13)
e E ad

Where g = acc. of gravity = 32.2 ft. sec’:

 

  

 

¥ = specific wt. of water = 62,5 lbs. ft°
“= elasticity of the water. ¢ = 42,000,000 lbs. tt?
= = elasticity of penstock material in lbs. et"
D = Dia. of penstock in ft.
d = thickness of penstock wall in ft.
The value of E is variable for the same material
and for different materials. The following average values

have been taken from "Kent's" Mechanical FEngineers' Pocket

Book and from the German engineers' hand-book "Hatte".

61
CG - (g)

9 ibs. tt

" gast iron E = 15,000,000 lbs. in.” = 2.16°109 1bs. 2%"

Por steel plate E-= 28,000,000 lbs. in.” = 4.032°10

" wooden staves E = 1,680,000 lbs. in.® =_ 2.42108 lbs .f£t*
If we substitute the numerical values for g, y and e
in formula (13) and, for the sake of brevity, place the

~l .
letter K —~B 101° we will obtain:

Bee EO (13-a)
23.5 — KeD

d

The value of K can now be computed for the various
penstock materials, and will be as follows:
Steel plates K = 0.232
Cast Iron K = 0.464
Wooden Staves K aw 41.50

(Formula ( 13 ) was first derived by Lorenzo Allievi,
C. E., of Rome, Italy, and published by him in 1903 in the
Italian paper "Annali della Societa degli Ingeneri ed
architetti"” under the title "Teoria generale del moto
perturbato dell' acqua nei tubi in pressione". In 1904
this article was translated into French under the title
"Theorie generale du mouvement varie de l'eau dans les
tuyaux de conduite™ in the French paper “Revue de Mecanique”. )

Based on the foregoing formula and values for K, the
atvached Table III giving: values of a has been computed,
and includes diameters of pipes from 1 ft. to 20 ft. with
thickness of walls from +" to 5".

For practical use, at least for preliminary purposes

or when the table is not at hand, the value of a can be
Eo
Cc - (h)

taken as 3,300 ft. per sec. for customary diameters and
thicknesses of penstocks.

The effect of the velocity of vibrations upon the
pressure variations depends upon the regulating time T,
as will be seen from the following discussion.

If we assume a penstock of length IL, as shown in the
accompanying sketch, Pigure 11, with a basin at its upper
end and a gate at the turbine, the water with the gate open
will have a velocity v. If the gate is closed we will
have a variation in the velocity, with corresponding
pressure variations and vibrations. As shown above, the
vibrations will travel along the pipe line with a velocity
a ani will reach the basin in a time t = Z, If the basin
is sufficiently large, that is, if its area is many times
that of the penstock, then the hydro-dynamic reaction, due
to the large body of water, is the cause of a new series of
vibrations which proceed down the penstock with the same

velocity a as before and the gate is again reached by them

in the time t = 24,

These reproduced waves coming from the basin will
interfere with the waves traveling upwards and have the

tendency to diminish the resulting pressure variations in

all sections of the penstock, therefore also those in the
gate area. This would indicate the following:

lst.- Up to the time t = =* (period t = 0----t = 24)

the pressure in the gate area is constantly increasing as if

63
c - (i)

the penstock were indefinitely long. See Figure 12.

end.- Starting from the moment t = £4 the pressure
variations are weakéned by the retroceding waves or
vibrations, and we conclude that the pressure must rest
constant from the time t = 22% until the gate stops in its
travel, (t = 1) as both series of vibrations have an equal
effect on the pressure.

Srd.- When the gate is stopped in its travel (t =T)
we get different conditions depending upon whether it is
fully or partially closed. If it is fully closed, fluc-
fuations will occur having a period of £4, but these will
become smaller and smaller until the original pressure is
reached.

If the gate area is only partially closed in the time
t =, the pressure rise will decrease asymptotically, since
the column of water still keeps moving, but with a changing

velocity.

The period from t—o to t= a+ only will be of
interest to us in this discussion, since during this period
the maximum pressure rise or drop occurs.

The graphical illustration given in Figure le explains

our reasoning more clearly. From the point t— =o to

t = =+ tne pressure is constantly increasing or decreasing.
From the point t = a to t —T the pressure remains

constant.

64
G - (3)

From the above we see at once that it would be working
acainst desired results should we choose a regulating time T
for a covernor less than the value a, without awaiting the
weakening effect of the reflected waves which arrive at the
cate in the time a,

Therefore the governor regulating time T should always
be greater than the time t = 24,

To show this fact, just stated, more clearly, we will

consider both cases: (a) 7 3 ae end (b) T> a
As we have already shown, if the gate is closed in a

= zt the pressure rises as long as the gate is kept

time
moving, and if the gate is totally closed we will obtain
the maximum possible pressure obtainable in a penstock.

This maximum pressure can be computed by means of the
formule: Ene, = Ho + == (14)
where:

Ho = initial head.

VY = velocity in the penstock.

££ -— acceleration of gravity.

This formula shows us that every change in velocity of

one ft. means an increase or decrease in head of

 

axl _5,300x1 about 100 ft. if S22,
g 32.2 = *

From the example which we have considered under

formula (12) we have: vp — Vv, = 5 ft. H=100 ft. L = 500 ft.

2 2x 500 3.300 x 5
n omen — e ° = <2 —
a “3,300 = 0.3 sec Hnax .= 100 +— BO.

65

 

All
C - (k)

600 ft., which indicates that the pressure would rise from

100 ft. to 600 ft. if we closed the gate completely in the

time t = zu = 0.5 sec.

&

The time t = £4 therefore represents a critical time,
which should always be calculated, and the governor regulating
time chosen as much greater as other conditions will permit.

If we now compare formula (9), originally derived for
a maximum pressure rise, with formula (14), we obtain a

difference for Hngy. as follows:

From formula (9) {D E) “50 OSS Ss 5 = about 2.6 = 260%

 

(DH) = 260 ft.
Hnax, = 100 + 260 = 360 ft., a8 compared with 600 ft.
as obtained from formula (14). This clearly illustrates
the importance of considering the effect of the vibrations
and elasticity of the water as well as the elasticity of the
penstock walls.

It is evident that the theory by means of which formula
(14) is derived shows the maximum value of H, as for the
period t= =Q to ft = £4 (2 wa 2) it is immaterial in
what time the kinetic energy is stored in the penstock. If
we close the gate completely the same energy is stored up,
whether the gate is closed in 0.1 sec. or 0.3 sec.

As has already been shown, for a closing time p>&t

a
the pressure rise is constant, and this constant pressure
increase or decrease represents the maximum possible value

for such a regulating time, and may be calculated by means

66
 
Cc - (1)

of the following formula:

{pH} = 2(n Jn? + 4) (15)
true for the period ft = it —-- tT
where {DH} = pressure variation (Pw) x 100 =

pressure variation in %)

 

n =m VY. _ (formula 9)
gHT
Use + sign for closing gates.
1t — 1 1" opening w
This formula (15) is most important in our calculations,

since it gives us the momentary head under which a turbine
must operate during a load change for any given regulating
time.

The results of this formula plotted graphically give

us two curves, one for increase of head and one for decrease

of head.

(DH)

H

For a decrease, if n= infinity, {DH} will asymptotically

For an increase, if n = infinity, = infinity.
reach the value (1) one. See Figure 13.

It will be noted that formula (15) contains just those
factors which we missed in formula (9). According to
formula (9) our results, if plotted, would appear along the
straight line as shown, practically midways between the two

 

Curves, + OH) and — oF) . See Figure 13.

We further note thet for a regulating time greater than

n= and variations of pressure not exceeding 20 to 25%,

67
 

 
C - (m)

formula (9) may be used approximately. For larger variations

(D H) (D H)
q and "a

siderable, and formula (15) must be used even for approximate

the difference between + becomes con-

results.
The attached Tebles IV to IX have been computed from

formula (15) and are based on different values of = (where

L is the length of the penstock and H the effective head)

and for velocities up to 20 ft., each table giving the

pressure variations, both + z H

for a given
regulating time ®.

These tables will be found of sreat benefit to quickly
determine the rezulating time and the corresponding pressure
variations for different penstock diameters, providing I
and H are given, as they are in practically every case.

But in determining or assuming the regulating time, it must

be remembered that it always must be greater than ay

&
Referring to the tables and the case already used

 

to illustrate, nanely: L = 500 ft., H = 100 ft.,
v=65 ft. and? $2. sec.,

LZ 500-_ (D H) a
z= 100 = 5. a - + 0.471 and — 0.352 = 47.1 and 32%.
From formula (9) we would obtain oH) = * 39% as the

approximate average value of the exact results. The

tables permit of interpolation.
It now becomes necessary to analyze the effect of the

pressure variations on the speed regulation.

68
CGC - (n)

As already stated, if the regulating time f is chosen
ereater than 2+ the pressure rise or drop, and therefore the
head remains constant and its maximum value may be calculated
by fomzula (15) according to which the tabular values are
found. Ye may therefore assume that the turbine is operated
at a constant head during any load change, the head varying
for different per cent. of changes, and the head for any
part load cnange may be found by substituting the new pen-
stock velocity in formula (15) or from the tables direct.

From formula (32) we see that a is directly proportional
to N (= H.P.) the load. Therefore if the head is varied
the capacity varies as H (H.P. varies as the square root
of the head cubed), and for a new head = (H+(DH) )
the capacity of the turbine rises with (z + (D z)) 3/2 and
the new capacity of the turbine is s(x + (D 2)? , which
is the case when load is thrown off.

When the load is thrown on the available energy decreases

with the term

 

— 4
(H — (D H))%/e
To obtain the results in per cent these terms are

3 |
transformed as follows: (2 + (2) . for closing gates,

 

 

and ( S A) » SO that finally we get a speed variation
1—
H )

dad modified by pressure variations + a8 given by the

 

following formulae:

69
Cc - (o)

For closing gates (Load thrown off) d™ a= dere, (a + omy” (16)
feft. (17)

(2 - (D ODyr

 

" opening gates (Load thrown on) a=

After making a number of calculations according to these
formulas, it will be noted that the results from both are
practically identical within certain limits. That they are
not identical throughout the range of the pressure variation

tables can be seen from the following:

 

 

deff.
For — (DE) 1.00 (100%) a™= : = infinity, whereas
1-1)*/2
| 3/,
for + ow) = 1.00 a = dere. (1 + 1) 2 = a definite value.
This shows us that, up to a point where a = + l;

the results for d™ are practically identical for opening or

closing the gates. Then the a@™ ourve turns up for opening

_ (DE)
H

curve for closing gates goes on undisturbed, as shown by the

and becomes infinity with = 1.00, while the a
accompanying diagram, Figure 14.

This indicates that for practical problems, where
— DE) wou1a never be allowed to reach the value of 1.00,
it is not necessary to calculate d™ for both opening and

closing gates, but that either one of the formulas (16) and

(D H)
H

100%, a" for opening gates becomes considerably greater than

 

(17) will give the desired results. As — approaches

a™ for closing gates. In such cases it is well to calculate

70
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a’ for both movements, particularly as frequently it is not
necessary that d"' for opening’ gates be as low as a" for

closing gates.

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fcr Opening or Closing Lime of 1 Second.
Sign indicates Pre ssure LT¢s@.

M TOP...
pet results in percent melziply Cabucar values hy 100.

Velocities tm Leustock | 2H SEER per. econ.

(Fe |2P¢(3Fe |art|oFzloPt |7Fel/QFtl[OFt OF ties HPL CRU. WBE 2. zee |
+ 0.032 |0.064| 0.098 [0.132 [0.169 [0.205 5 [owas |0.283 0.322 10 $72 j0-4 50 0.545. eaeio. 740 6.850] ¢

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OK (0045 |0.083)0. ISI oues ‘0.206 lozagioz79| 0.309 |0.342|/6.391 jo: 2S C4713 O52RLU SA O59

‘9 + 10.065 [ONSR ORO7 | 10.280 |0.360 !0-45010 543. OSs 0.740 /0.840} 107 ‘1.33 lor 1.9l e222 +
in '0.0€0 CMG IOI! lORI9 lores fou | O35 0388 0425 10456! O.S1S. 0.875 0.618 (0655, OC»

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- ‘0.075 O43 ORS | 10.265 [caee 0370 C415, 0460 0497/0534 0.593 0.642. 0691 ‘0.727 Ot Te :

1.08 ‘126 47 1.31 24a ‘2.99 3.60 4. 4 | t

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o> '0 090 lO. 169 |o 024410 310 10368 0.425 OAT! 0.5/8 0.56010.598 0655/0 705 0.74 & 0.780 OBIS -
33 1.55 |1.B6 Re-7i i306 GBI 66 15.60 | 4

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3 _ 0.104 10.193 10.276 | 0.550 [ess | C28 19) 1.52 80. 58¢ Ce1e oes: 3 sle7c7 \orse € aw t G22 € e5 on
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= C130 12410 343 012510495 0.556. O02 06950032 0128S 0.780 10 B22 O85 OLE cou
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oo FORE E NOW Ze.) 59.7 ol.o &F.5 JkOO (od 0 198.6 RAGS 346.0 -4 15.06 G22. T8900 370 t

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CO - «, IC 2.98 6.998 nce $.0C ' EOC. 4,00 1.00 moO f.0¢ I ppoo | Uo, $0 1.OO 1.06 7
joo. t.'* iSO ‘3.6 B3.5 1558 240.0 S41 “1 16.0 6200 185.0 96 TO 1395.0 19000 2496.0 314 3 Ic.a +

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— ~ ———_ —— - —_— he eee - ~* _ -_—_
FRE SSURE VARIA LIC INS,
bor Opening or Closing | L ume cf R. Seconds

t Sign cuadtcates Pressure Rise.

——_ ” ¢ Dro
ee To get results vt Per cent multiply tabular values by ICQ.

L — Velocitres Ln”. Pensteck in . feet per. Second.
A IEE ZER. St AEE SEY GE t7Et BELIEt. OFt I2Ft IFeloEt, EU KOFE,

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a 0.016 ‘0.032 0.047 0.0600 075 0.090 ‘0104 (O1G 0.130 'O 144 017? 0195 CREO OPA OR TA -
+ acest 0.047 0.073 0.096 0.123 0.150 O1GS 0.205 'ORS# 0 ZE0'O. 0322 ‘C0 3B€ CF46 0.520 © 587 t

~ |e. CR-t U.C# A 0.068 (0038 0.109 ‘0. (Bt 0.142 ‘0169 0.189 0.206 10.244 +0279 (0309 (0.342 "0391, :
+10.03 3: 0.069 0.09 & O13R 0./69 0.207 0.243 0.280 03R2 0360: 0-450 0 543 0638 0 740 OBto +
i 0.032 0.060 0.090 0.116 0.146 O17 0195S 0219 0244 ORES | 0-311 0351 0388 0.425 0-156 -
+. C.0F 1 0.081 O1RB 0168 0.214 0.2 700.310 0.361 O46 0470 0 590. 0.716 0.850 0.990 | 140 t

on
* -'o. O39 0075 0.109 0143 0. 176 CRIS OR3T7 ORCS 0.2940 320, 0370 Ons OAGO 04970 534 -
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_~ ,0.047.0.090 0.130 0.169 0.209 O24 ORTT O.319 0 BFR 0.368 OARS OFT! 0.518 (0.560 0 598 -
t iC 058 OIG 01/78 0243 asi 23850460 0.544 0.625 0.708 | 50-90 b110 1330 1950 1860 +
— 054 0104 ‘ALsl G19S 0239 0276 03S” 0.350. 0.385 0415. OFTS0.S528 | 0.580 0.6/2 0655 —
+ 0.066 C132 0.205 028403064 (0-453 OS4FDO63S ‘0.740 0.840 1070 1.330 1620 1910 %260 +
—|0.0GR O-1IG ‘0.1 Vi (0 R20 0.266 0312 0.350 0.386 0-426 ‘OF 5C'0 S522 0572 0.615 0.655 0693,
“410.073 O.150 0.233 C323 ‘0.416 0.5200 6230 736 0.B6O 0.985. | 260 1.570 1.910 22RBO Z6IC t
10068 0.130 0.189 0 244 0294 0.543 0385 0425 0-463 0495 0.556. OGIR 0655 0692 O7¢8 |
+: 0.08 0168 0261 0364 OFT! 0.592 0 716 ost9, 0988 11390 | +400 1BGO 2 240.2 690 3 190 t
~\a075 014A 0.209 OR6 (0.320 O37 ONS (046010496 CEXT 0.595.0654 0693 ‘0. 1300758
"40.098. 0.206 asee OA4350 ‘0.590: ‘0.739 0 90 | O80: 1.260 | 460 1.90 at 2.990 3.60 ~f 310 +
'=10.090!0.171 0.244 0.310 0360 OARS OATS'A515 0.5580. 59O06250.712 0.758 0.1820.812 —
ti0.1eG (0.2-43 0385 0 $40 0.715 '0 902 i1o 1.320 (158 1.82 R-tO 3.06 BBR : 4.060 560 +
-\o 0.104: 0195 0277 ‘0350, 0420! ‘OFTBOS25.0570 'OGI2 0.6400 H12 0.746 0.792 2 0. BER O eto —
+ OLS 0.285 0-450 0.640 (0848 1.080 13S 1.6201.910 R.RF 297 384 AB 588 Tk
~ O18 | O22 0.512 0.390 0-460, ASIC 057006200655 0.695" 'O7S3 0790 0830 O&6l aB%mW
+ 10.450 03221052 | 0.74 2.99 N27 1ST LOR 228 Z.€6 3.60 “4.72 5.87 7.22 Bis +
_ 0.1430 O.RF4A O342 CABS 0496 0.5600 O/R 0660 0.693 0.730 | 0.780) 0.820 0 B60 C.BB0.0.920
toned | 0.375 0.594'0850 L140 hate Le4+O 2230 2.69 3. \GO -7.2 7 seo 710 875 19 60
| 10.144 O8275 0.372 O60 | 0.533 °. 595) 0.6 5¢ Uk VS 0.730 0.760 ‘0.810 0.8-40'0.BB6 6.895 6.95C.
! 1S. “+ 0.264 0.595 0.99 14 2.02 2.70 | ‘3.13¢ 4260 525 6.2808 13 11.70 ‘It BO 18.6 PL +
7/0. RAO 0.374 0497 0.595 ness 10.735 ; 0.759. CEIE O.B40 ‘0.B6C '0898 0.9300 945 ©. 35C O9EC
! 20. + 0.3G)1 0.B5° 146 & RA IAG [+ Si (3.60 © “1.020 &. 190° 10.60 14 80, 19. JO. 1&9. 72 ‘32. 2C (SOEC +
| - 0.265 0461 « 10.60 069 0. 16 jOBb2 0.84 0.87 6895 C915 0.930 0.9406. 210 | C.98C Lee =
25, “+ O.F73 I. 140) Z.04 3.20 4.58: 6.35 830 10 50 13. RO IG. ecto d60 30.60 39. a 50.00, GI ic +

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so. +t 0.59 147 2.69 4.301630 B75 LTO 14.80 IBGO 22.6 32 10 -4-4,0 5.00 NSO Be
i 0.371 0.60 073 08! '0.853'0910 0.950:0955 0.960099 '099 100 1.00 Lee bee
ig "+ 0852 220 1432 7.10 ‘10.70 {15.00 19.90 2.5.30 32.2 'BI4O1SE-10 74.2 1OC.C17 7.0 158.0 t
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“+ "2.260: TRO ‘1490. 258 "39.6015 7 00 77 2100.0! 126.0" 155.0 223.0: 1306 0396.0 505.0 6% 00 +
205" (0.6295 0.90 10.98 5 |0.98 0.995 0995)1.00 | | }.00 |.00- 1Oo ‘yeolle OO 1.09 4.00] Gc -

}O0. MY 3.19 060 22 90 S86 \6rseises | 120.0 155.0 197.0 240.0 34704 16.062 5.0°785.0'9670 +
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Foy Openiny or Ulosing Lime of 3 Jeconas.

t Szyn _zwandticau tes Pressure Feiuse.

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